PARAMETER SENSITIVITY ANALYSIS OF PHOTON RECYCLING IN GALIUM ARSENIDE SOLAR CELLS: METHODOLOGICAL DEVELOPMENT

1. PARAMETER SENSITIVITY ANALYSIS OF PHOTON RECYCLING IN GALIUM ARSENIDE SOLAR CELLS: METHODOLOGICAL DEVELOPMENT GRACE CAREY, ILYA KORSUNSKY, ARJUNEN KUTAYIAH, KATHLEEN MCGOVERN, LAUREN SWADDELL

2. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

4. U.S. Energy Consumption and Production predictions Source: U.S. Energy Information Administration,Annual Energy Outlook 2011, Early Release, December 16, 2011.

5. U.S. energy consumption in 2009

6. U.S. Primary Energy Flow

7. Source: U.S. Department of Energy, Department of Fossil Fuels, 2011

8. Carbon Dioxide Emissions

10. Ice Core Data and The Keeling Curve

11. Vostok Ice-Core Data

12. Alternatives to fossil fuels? The suspense is terrible… I hope it’ll last

13. Nuclear Energy 400 nuclear plants in the world 100 nuclear plants in the US alone Powers ~15% of US energy needs Relies on the use of uranium and other fissible materials to generate electricity Uranium is a finite mineral resource Cooling methods often employ the use of local water systems endangering aquatic life Nuclear power plants in the US produce 2000 metric tons of radioactive waste Nuclear disasters can emit large amounts of radiation which can be lethal and detrimental to the environment

14. Solar Energy Sustainable and renewable resource which does not emit greenhouse gases ~1% of U.S. energy Solar energy hitting the earth is approximately 274 million giga-watt/year = 8.2 million quads of Btu/year Solar cells currently have an average efficiency of 15%  369 thousand quads of Btu/year can be collected if all land mass of earth had solar panels Total potential for solar energy is 444,000 TWh The world’s total energy consumption is 132,000 TWh The total annual energy consumption in the US is less than 0.5% the theoretical amount of sunlight received

15. Solar Cell Efficiency Tables

16. Solar Cell Efficiency Tables

17. Solar Cell Efficiency Tables

18. Solar Cell Efficiency Tables

19. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

20. N type Electric field Space Charged Region P type N type P type Contacts Photon N type P type

21. Radiative Recombination Conducing Band, Ec Band Gap, Eg= Ec - Ev Valence Band, Ev

22. Photon Recycling • Re-absorption of photons generated in a semiconductor device as a product of radiativerecombinations. • Increases efficiency by 1-2% • GaAs • Semiconductor • Direct Band Gap • No energy is lost to phonons (lattice vibrations) as a result of radiativerecombinations. • Good for optical devices

23. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

24. Modeling: Motivation • Goal: create the best solar cell we can! • Efficacy • Cost • Environmental Impact • Need some design guidelines • Computational model handles complexity

25. The Model

26. The Model • Output: Photon Recycling Rate • Inputs: • Temperature • Front Surface Reflection • Width • Angle of Incidence • Refractive index • Light Wavelength • Internal Surface Reflectance • Reflectance of Metal Grid • Front Internal Shadow Factor

27. How do we use the model? • Optimize Photon Recycling over the input parameters

28. Dealing with Complexity • 9 input parameters => 9 dimensional hypercube • Are all the parameters important? • Sensitivity analysis gives importance of each parameter • Cut down search space

29. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

30. The simplest and most powerful relationship between independent and dependent variables is linear. The dependent variable can be predicted from the independent variable by fitting the data to as follows:

31. The problem is almost always more complicated. If the dependent variable is a function of multiple independent variables, we have: This describes multilinear dependencies for only one sample; for n samples y can be written as a column vector and the values of x form the rows of matrix X:

32. In multiple linear regression, the solution for the b vector take the form: Can anyone see a potential problem here?  The formula for b depends on the invertability of the product matrix of the X row vector and the X matrix!

33. Partial least squares (PLS) regression does not depend on the invertability of input data. Assume a linear relationship between independent parameter matrix X and dependent output matrix Y: PLS regression uses a variation of the NIPALS algorithm to find the best approximation of this linear relationship in the form of regression matrix, B.

34. What does the PLS algorithm look like?

35. The Model

36. The previous complexity can be reduced to the following:

37. The regression coefficients (Bpls) can give us the following information: • (1) Significance of independent parameters to output(s) of interest (2) Prediction of dependent parameters from independent parameters (unlike PCA) (3) Indication of parameters to be tested in future experiments • (4) Unreasonable results indicate that a mathematical model needs to be reevaluated in some regard

38. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

39. The Model

40. Photon recycling rate (GPR): function map KEY Experimental variables ∞ 1 Constants EG -1 Functions ∞ GPR(x) = 2π ∫d E∫ dμα b(E, x, μ) 0 b(E,x, μ) = w x ˆ bn(E,x) = α = Φ= exp - RF RR μ = cosθ ΨOF = RR ∫bn exp ∫ bn exp dx’ dx’ + exp E = ΨOR = RF∫bn exp ∫bn exp dx’ + dx’ RF = κF* FSF + ρF * (1 – FSF)

41. Photon recycling rate (GPR): function dependency chart GPR E b μ α Φ RF μ α ΨOF ΨOR bn α μ RF bn μ α RF bn E E E

42. Photon recycling rate (GPR): code sample

43. Outline • Motivation: Environmental Impact • Solar Cells: Behind the Physics • Modeling and Optimization • Sensitivity Analysis (PLS regression) • Implementation • Results • Design Conclusions • Future Directions

44. We can apply the PLS algorithm to our input and output data. Input Output 1.5 Trials 0 -1.5 q k l r T k W F G F SF hat n P R Parameters

45. Results of PLS regression: B Output Input Matrix P L S • k 1.5 • l • kF • r = * • FSF 0 • W • q • nhat -1.5 T q k l r T k W G G F F SF hat n P R P R

46. Results of PLS Regression: G Regression Coefficients P R 1 These give quantitative insight into how changing input parameters affects output. 0.5 0 Significant parameters include wavelength of light, temperature, and the front reflectance. -0.5 -1 q k l r T k W F F SF hat n

47. Accuracy of regression

48. Conclusions: PLS regression is an accurate tool for both determining parameter sensitivities from our simulated data sets and predicting the output variable data of interest. As conserving energy is of optimal interest to the environment, photon recycling is an important physical phenomenon to energy conservation and solar cell efficiency. From our regression analysis, the parameters which should be maximized in future cell design are wavelength of light directed at the solar cell, temperature, and front reflectance.

49. Future Directions • Function for cost • Function for environmental impact • Convex optimization

50. Questions • Any? • No? • Thanks!